On heating blue coloured powder of copper (II) nitrate in a boiling tube copper oxide (black), oxygen gas and a brown gas X is formed. Which type of reaction is this? 

The given reaction is a thermal decomposition of copper nitrate. On heating, the compound gains enough energy to break the bond between them and breaks into smaller compounds. The brown gas produced during the reaction is nitrogen dioxide.

According to the query, when heated in a boiling tube, copper nitrate separates into brown gas, oxygen, and copper oxide. The reaction can be written as-
$2\mathrm{Cu}{\left({\mathrm{NO}}_3\right)}_{2\left(\mathrm{s}\right)}\stackrel{∆}{\to } {\mathrm{CuO}}_{\left(\mathrm{s}\right)}+{\mathrm{O}}_{2\left(\mathrm{g}\right)}+4{\mathrm{NO}}_{2\left(\mathrm{g}\right)}$
Burning copper nitrate in a boiling tube causes it to split into its simpler components. For this process to finish, some heat is necessary. The decomposition reaction satisfies the requirement that a single component dissociates into two or more pieces in the presence of heat. When heat is present, this process falls under the category of a thermal breakdown reaction.

Hence, the correct answer is option A.